Loops in a Platonic solid

Question

For a given Platonic solid, how many closed paths are there on the edges of the solid if each edge can only be traversed once, and paths related by a rotation of the solid are considered the same?

For a tetrahedron, the answer is 2 (a triangular path or a path that touches each vertex once). For a cube, the answer is 4 (a square, an elbow, a "U-box", or a zig-zag). Haven't done the others yet...

Answer 1

For an octahedron, it's 5, since there is only one path for 1,2,3 and two different paths for 4.

For the dodecahedron 1 and 2 just the one 3 two different paths.
4 In a row, 2 (one pole to pole, and one other), one with the triangle, and one with a quadralateral, or 4 together. 5.